Nature of intrinsic relation between Bloch-band tunneling and modulational instability

On examples of Bose-Einstein condensates embedded in two-dimensional optical lattices we show that in nonlinear periodic systems modulational instability and inter-band tunneling are intrinsically related phenomena. By direct numerical simulations we found that tunneling results in attenuation or enhancement of instability. On the other hand, instability results in asymmetric nonlinear tunneling. The effect strongly depends on the band gap structure and it is especially significant in the case of the resonant tunneling. The symmetry of the coherent structures emerging from the instability reflects the symmetry of both the stable and the unstable states between which the tunneling occurs. Our results provide an evidence of profound effect of the band structure on superfluid-insulator transition.